The following two transfer functions look similar. The difference is that their numerators and denominators are reversed. One is a tuned (narrow bandpass filter), the other is a notch (narrow bandstop) filter. Their frequency responses seem identical except for one being a notch and the other tuned. However, their step responses are quite dissimilar.

\[
\begin{aligned}
& T_{1}(s)=\frac{s^{2}+50 s+50000}{s^{2}+500 s+50000} \\
& T_{2}(s)=\frac{s^{2}+500 s+50000}{s^{2}+50 s+50000}
\end{aligned}
\]

(a) Plot the frequency response of each function. Which is the tuned filter and which is the notch filter? What is the height of the tuned filter and the depth of the notch filter? Are their center frequencies the same?

(b) Plot the step response of each. Which is underdamped and which is overdamped? For the underdamped case, find \(\alpha\) and \(\beta\) from the plot. For the overdamped case, estimate the dominant exponential from the plot.

(c) Locate the critical points of the numerator and the denominator of each transfer function. Relate the critical points found in the step responses with the critical points found in factoring the transfer functions. Explain why one is overdamped and the other underdamped.

(d) Suppose that the numerator and denominator in each transfer function were the same. What would the frequency response be?